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A tridiagonal matrix is a square matrix with zeroes everywhere except on the main diagonal and the diagonals just above and below the main diagonal.

Let $T$ be a real anti-symmetric tridiagonal matrix with elements $t_{12}=c_{1}, t_{23}=c_{2}, \ldots$, $t_{n-1 n}=c_{n-1}$. If $n$ is even, compute det $T$.
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For an anti-symmetric matrix, the determinant is equal to the square of the magnitude of any of its eigenvalues. Since all eigenvalues of an anti-symmetric matrix are imaginary, the determinant of an anti-symmetric matrix is always positive. The determinant of a tridiagonal anti-symmetric matrix can be calculated as the product of the diagonal elements, which are the squares of the eigenvalues. Therefore, if $n$ is even, the determinant of $T$ is equal to the product of the squares of the eigenvalues, and is positive.
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